For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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14
votes
2answers
461 views

Geometric interpretation of the cofactor expansion theorem

I find the geometric interpretation of determinants to be really intuitive - they are the "area" created by the column vectors of the matrix. Could someone give me a geometric interpretation of the ...
14
votes
2answers
2k views

Integral of matrix exponential

Let $A$ be an $n \times n$ matrix. Then the solution of the initial value problem \begin{align*} \dot{x}(t) = A x(t), \quad x(0) = x_0 \end{align*} is given by $x(t) = \mathrm{e}^{At} x_0$. I am ...
14
votes
2answers
3k views

Topology of matrices

1.Consider the set of all $n×n$ matrices with real entries as the space $\mathbb R^{n^2}$ . Which of the following sets are compact? (a) The set of all orthogonal matrices. (b) The set of all ...
14
votes
1answer
960 views

Max dimension of a subspace of singular $n\times n$ matrices

I am sure the answer to this is (kind of) well known. I've searched the web and the site for a proof and found nothing, and if this is a duplicate, I'm sorry. The following question was given in a ...
14
votes
2answers
263 views

Why does calculating matrix inverses, roots, etc. using the spectrum of a matrix work?

Suppose $A$ is a $n \times n$ matrix from $M_n(\mathbb{C})$ with eigenvalues $\lambda_1, \ldots, \lambda_s$. Let $$m(\lambda) = (\lambda - \lambda_1)^{m_1} \ldots (\lambda - \lambda_s)^{m_s}$$ be the ...
14
votes
1answer
257 views

What do characteristic polynomials characterize?

Let $R$ be an integral domain and $F$ a finitely generated free module over $R$. For a linear transformation $\alpha\in\operatorname{End}_R(F)$, the characteristic polynomial is \begin{equation} ...
13
votes
8answers
2k views

I need to calculate $x^{50}$ [duplicate]

$x=\begin{pmatrix}1&0&0\\1&0&1\\0&1&0\end{pmatrix}$, I need to calculate $x^{50}$ Could anyone tell me how to proceed? Thank you.
13
votes
2answers
689 views

What's the name for the property of a function $f$ that means $f(f(x))=x$?

I can think of several examples of functions such that twice application of the function is equivalent to no application of it. Additive inverse Multiplicative inverse Fourier transform Complex ...
13
votes
4answers
690 views

Expected Value of a Determinant

Suppose that I construct an $n \times n$ matrix $A$ such that each entry of $A$ is a random integer in the range $[1, \, n]$. I'd like to calculate the expected value of $\det(A)$. My conjecture is ...
13
votes
2answers
33k views

Transpose of inverse vs inverse of transpose

I can't seem to find the answer to this using Google. Is the transpose of the inverse of a square matrix the same as the inverse of the transpose of that same matrix? Thanks!
13
votes
4answers
308 views

Is it true that all matrices in $M_2(\mathbb R)$ is the sum of two squares?

I recently show that every polynomial with real coefficient and $P$ is always positive. is a sum of two squares of polynomials. These questions also appear often in arithmetic. What if we change ...
13
votes
2answers
563 views

Characteristic polynomials exhaust all monic polynomials?

Let $A$ be an $n\times n$ matrix, then $\mathrm{char}_A(x):=\det(xI-A)$ is a monic polynomial of degree $n$. It is called the characteristic polynomial of $A$. My question is the converse: Let ...
13
votes
4answers
460 views

How to prove that $A^2$ is a symmetric matrix

Conjecture 1 : Let $A$ be a real matrix such that $A^5=A A^T A A^T A$. Then $A^2$ is a symmetric matrix. (here $A^T$ denotes the transpose of a matrix A). I guess that the following is also ...
13
votes
2answers
2k views

$AB-BA=I$ having no solutions

The question is from Artin's Algebra. If $A$ and $B$ are two square matrices with real entries, show that $AB-BA=I$ has no solutions. I have no idea on how to tackle this question. I tried block ...
13
votes
3answers
231 views

If $A$ is positive definite, then $\int_{\mathbb{R}^n}\mathrm{e}^{-\langle Ax,x\rangle}\text{d}x=\left|\det\left({\pi}^{-1}A\right)\right|^{-1/2}$

Let $A$ be a positive definite real $n\times n$ matrix. How can I prove that $$ \int_{\mathbb{R}^n}\mathrm{e}^{-\langle ...
13
votes
4answers
399 views

Exponential lower bound for the determiant of a (0,1)-matrix

Give matrices, which only contain 0 and 1, and their determinant grows exponentially. In other words, show an $n \times n$ matrix for all n, which only contains 0 and 1, and $$\det A(n)>d \cdot ...
13
votes
4answers
494 views

Choosing an orthonormal basis in which a linear operator has a sparse matrix

Given a linear operator $T$ on an $n$-dimensional vector space $V$ (over $\mathbb R$), I want to find an orthonormal basis for $V$ in which the matrix of $T$ is sparse (has many zeros). How many zeros ...
13
votes
3answers
598 views

Let $A$ and $B$ be $n \times n$ real matrices such that $AB=BA=0$ and $A+B$ is invertible

I came across the following problem that says: Let $A$ and $B$ be $n \times n$ real matrices such that $AB=BA=0$ and $A+B$ is invertible. Then how can I prove the following: rank $A$+ ...
13
votes
3answers
460 views

If $C$ commutes with certain matrices $A$ and $B$, why is $C$ a scalar multiple of the identity?

I'm self studying Steven Roman's Advanced Linear Algebra, and this is problem 10 of Chapter 8. Let $A,B\in M_2(\mathbb{C})$, $A^2=B^3=I$, $ABA=B^{-1}$, but $A\neq I$ and $B\neq I$. If $C\in ...
13
votes
3answers
871 views

How to show determinant of a specific matrix is nonnegative

How to show that $$\det A= \det \begin{pmatrix}\cos\frac{\pi}{n}&-\frac{\cos\theta_1}{2}&0&0&\cdots&0&-\frac{\cos\theta_n}{2} ...
13
votes
1answer
2k views

The Determinant of a Sum of Matrices

Given $N$ $n \times n$ matrices $\mathsf{A}^{1}, \dots, \mathsf{A}^{N}$, \begin{align} \det \left( \sum_{i = 1}^{N} \mathsf{A}^{i} \right) = \sum_{\sigma \in S} \det \mathsf{A}^{\sigma}, \end{align} ...
13
votes
1answer
199 views

Compute $\det(A^n+B^n)$

Let $A, B $ be two real $3\times 3 $ matrices, $AB=BA$, and $ \det(A-B)=\det(A^2+B^2)=1,\det(A+B)=3, \det(B)=0 $, then, what is ? $$\det(A^n+B^n)$$ here $n$ is a positive integer. The problem ...
13
votes
3answers
454 views

when does $\det(AB^T+BA^T)\le \det(AA^T+BB^T)$ hold?

When does the following matrix inequality hold? $$\det(AB+B^TA^T)\le \det(AA^T+BB^T)$$ $A$ and $B$ are any real matrices. My reply gives a counter example. The question is under what condition ...
13
votes
3answers
299 views

Prove the inequality: $\prod_{j=1}^ka_{jj}\leq\left(\frac{1}{k}\sum_{j=1}^k\lambda_j\right)^k.$

To all those who are eagerly awaiting a new question, all those who love math, I give this challenge and I hope for you good moments of reflection. Let $A=(a_{ij})_n$ a real nonnegative symmetric ...
13
votes
0answers
227 views

How do I find the common invariant subspaces of a span of matrices?

Let $G_1, \ldots, G_n$ be a set of $m\times m$ linearly-independent complex matrices. Let $\mathcal{G} = \operatorname{span}\left\{ G_1, \ldots , G_n\right\}$ be the vector space that spans the set ...
13
votes
1answer
295 views

determinant of a standard magic square

What is the lowest positive, what the highest possible value for the determinant of a standard-magic-square-matrix of order n ? Are there singular standard-magic-square-matrices of any order ...
13
votes
0answers
188 views

Fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n, \mathbb Z)$

What is a simple description of a fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n,\mathbb Z)$? $\operatorname{GL}(n,\mathbb R)$ is the group of all real ...
12
votes
6answers
971 views

A matrix satisfying $AB-BA=B$

If $A$ and $B$ are two matrices of $\mathcal{M}_n(\mathbb{R}$) such that $$AB-BA=B$$ how can we prove that $B$ isn't invertible? my attempt: I found that $\mathrm{tr}(B)=0$ but I know that this is ...
12
votes
3answers
2k views

Why do the $n \times n$ non-singular matrices form an “open” set?

Why is the set of $n\times n$ real, non-singular matrices an  open subset of the set of all $n\times n$ real matrices? I don't quite understand what "open" means in this context. Thank you.
12
votes
4answers
918 views

Determinant of a specific circulant matrix, $A_n$

Let $$A_2 = \left[ \begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right]$$ $$A_3 = \left[ \begin{array}{ccc} 0 & 1 & 1\\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{array}\right]$$ ...
12
votes
5answers
318 views

Prove that $A^k = 0 $ iff $A^2 = 0$

Let $A$ be a $ 2 \times 2 $ matrix and a positive integer $k \geq 2$. Prove that $A^k = 0 $ iff $A^2 = 0$. I can make it to do this exercise if I have $ \det (A^k) = (\det A)^k $. But this ...
12
votes
3answers
18k views

A matrix and its transpose have the same set of eigenvalues

Let $ \sigma(A)$ be the set of all eigenvalues of $A$. Show that $ \sigma(A) = \sigma(A^T)$ where $A^T$ is the transposed matrix of $A$.
12
votes
4answers
5k views

How to calculate the matrix exponential explicitly for a matrix which isn't diagonalizable?

How can I compute an expression for $(\exp(Qt))_{i,j}$ for some fixed $i, j$ and matrix $Q$? When $Q$ is diagonalizable, we can diagonalize, but what can be done otherwise? Thanks.
12
votes
3answers
2k views

Recovering the two SU(2) matrices from SO(4) matrix

Since there is a 2-1 homomorphism from SU(2)xSU(2) to SO(4) there should be a way to recover the two SU(2) matrices given an SO(4) matrix. I believe I could set this up as a system of equations ...
12
votes
3answers
7k views

Can a real symmetric matrix have complex eigenvectors?

A Hermitian matrix always has real eigenvalues and real or complex orthogonal eigenvectors. A real symmetric matrix is a special case of Hermitian matrices, so it too has orthogonal eigenvectors and ...
12
votes
7answers
5k views

Relation between Cholesky and SVD

When we have a symmetric matrix $A = LL^*$, we can obtain L using Cholesky decomposition of $A$ ($L^*$ is $L$ transposed). Can anyone tell me how we can get this same $L$ using SVD or Eigen ...
12
votes
2answers
936 views

Why are the eigenvalues of these “bitwise XOR matrices” integers?

In the course of playing around with this question, I have hit upon a question of my own. Consider the $n\times n$ symmetric matrix $\mathbf X$ whose entries are given by ...
12
votes
3answers
319 views

Eigenvalue test faster than $O\left(n^3\right)$?

Given a real $n\times n$ matrix $A$, one can find the eigenvalues in $O\left(n^3\right)$ by using say, the $QR$ algorithm. Now, what if we guess an eigenvalue $\lambda_0$, and we want to know if it's ...
12
votes
2answers
719 views

Why is the identity the only symmetric $0$-$1$ matrix with all eigenvalues positive?

While thinking about this question I managed to convince myself that the identity is the only symmetric $0$-$1$ matrix with all eigenvalues positive. However, the argument is rather low-level. It ...
12
votes
2answers
590 views

How many $n\times m$ binary matrices are there, up to row and column permutations?

I'm interested in the number of binary matrices of a given size that are distinct with regard to row and column permutations. If $\sim$ is the equivalence relation on $n\times m$ binary matrices such ...
12
votes
6answers
3k views

Sylvester rank inequality

If $A$ and $B$ are two matrices of the same order $n$, then $$ \operatorname{rank} A + \operatorname{rank}B \leq \operatorname{rank} AB + n. $$ I don't know how to start proving this ...
12
votes
4answers
6k views

Is there a 3-dimensional “matrix” by “matrix” product?

Is it possible to multiply A[m,n,k] by B[p,q,r]? Does the regular matrix product have generalized form? I would appreciate it if you could help me to find out some tutorials online or mathematical ...
12
votes
1answer
4k views

Is it faster to multiply a matrix by its transpose than ordinary matrix multiplication?

I'm writing a program that multiples a matrix by its transpose, and was trying to find efficiency hacks I could exploit considering that the two matrices being multiplied are related. Any ideas?
12
votes
1answer
2k views

Probability that a random binary matrix is invertible?

What is the probability that a random $\{0,1\}$, $n \times n$ matrix is invertible? Assume the 0 and 1 are each present in an entry with probability $\frac{1}{2}$. Is there an explicit formula as a ...
12
votes
1answer
4k views

Relationship between eigendecomposition and singular value decomposition

Let $A \in \mathbb{R}^{n\times n}$ be a real symmetric matrix. Please help me clear up some confusion about the relationship between the singular value decomposition of $A$ and the eigen-decomposition ...
12
votes
2answers
628 views

Properties of 4 by 4 Matrices

Define $ A=\begin{pmatrix} x_1 & x_2 & 0 & 0\\ 0 &1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{pmatrix}, B=\begin{pmatrix} 1 & 0 & 0 & 0\\ x_3 ...
12
votes
1answer
1k views

Direct proof that nilpotent matrix has zero trace

Does anyone know a proof from the first principles that a nilpotent matrix has zero trace. No eigenvalues, no characteristic polynomials, just definition and basic facts about bases and matrices.
12
votes
1answer
14k views

orthogonal eigenvectors

I have a very simple question that can be stated without proof. Are all eigenvectors, of any matrix, always orthogonal? I am trying to understand Principal components and it is cruucial for me to see ...
12
votes
2answers
492 views

Product of nilpotent matrices.

Let $A$ and $B$ be $n \times n$ complex matrices and let $[A,B] = AB - BA$. Question: If $A , B$ and $[A,B]$ are all nilpotent matrices, is it necessarily true that $\operatorname{trace}(AB) ...
12
votes
1answer
113 views

Check membership in a matrix group

I'm looking for a (preferably somewhat efficient) algorithm for this problem: Given a normal subgroup of $SL(m, \mathbb{Z})$ generated by a finite set $\{M_1, M_2, \dotsc, M_n\}$, and some $A \in ...